| // Special functions -*- C++ -*- |
| |
| // Copyright (C) 2006, 2007, 2008, 2009, 2010 |
| // Free Software Foundation, Inc. |
| // |
| // This file is part of the GNU ISO C++ Library. This library is free |
| // software; you can redistribute it and/or modify it under the |
| // terms of the GNU General Public License as published by the |
| // Free Software Foundation; either version 3, or (at your option) |
| // any later version. |
| // |
| // This library is distributed in the hope that it will be useful, |
| // but WITHOUT ANY WARRANTY; without even the implied warranty of |
| // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
| // GNU General Public License for more details. |
| // |
| // Under Section 7 of GPL version 3, you are granted additional |
| // permissions described in the GCC Runtime Library Exception, version |
| // 3.1, as published by the Free Software Foundation. |
| |
| // You should have received a copy of the GNU General Public License and |
| // a copy of the GCC Runtime Library Exception along with this program; |
| // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see |
| // <http://www.gnu.org/licenses/>. |
| |
| /** @file tr1/modified_bessel_func.tcc |
| * This is an internal header file, included by other library headers. |
| * Do not attempt to use it directly. @headername{tr1/cmath} |
| */ |
| |
| // |
| // ISO C++ 14882 TR1: 5.2 Special functions |
| // |
| |
| // Written by Edward Smith-Rowland. |
| // |
| // References: |
| // (1) Handbook of Mathematical Functions, |
| // Ed. Milton Abramowitz and Irene A. Stegun, |
| // Dover Publications, |
| // Section 9, pp. 355-434, Section 10 pp. 435-478 |
| // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl |
| // (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky, |
| // W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992), |
| // 2nd ed, pp. 246-249. |
| |
| #ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC |
| #define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 1 |
| |
| #include "special_function_util.h" |
| |
| namespace std _GLIBCXX_VISIBILITY(default) |
| { |
| namespace tr1 |
| { |
| // [5.2] Special functions |
| |
| // Implementation-space details. |
| namespace __detail |
| { |
| _GLIBCXX_BEGIN_NAMESPACE_VERSION |
| |
| /** |
| * @brief Compute the modified Bessel functions @f$ I_\nu(x) @f$ and |
| * @f$ K_\nu(x) @f$ and their first derivatives |
| * @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively. |
| * These four functions are computed together for numerical |
| * stability. |
| * |
| * @param __nu The order of the Bessel functions. |
| * @param __x The argument of the Bessel functions. |
| * @param __Inu The output regular modified Bessel function. |
| * @param __Knu The output irregular modified Bessel function. |
| * @param __Ipnu The output derivative of the regular |
| * modified Bessel function. |
| * @param __Kpnu The output derivative of the irregular |
| * modified Bessel function. |
| */ |
| template <typename _Tp> |
| void |
| __bessel_ik(const _Tp __nu, const _Tp __x, |
| _Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu) |
| { |
| if (__x == _Tp(0)) |
| { |
| if (__nu == _Tp(0)) |
| { |
| __Inu = _Tp(1); |
| __Ipnu = _Tp(0); |
| } |
| else if (__nu == _Tp(1)) |
| { |
| __Inu = _Tp(0); |
| __Ipnu = _Tp(0.5L); |
| } |
| else |
| { |
| __Inu = _Tp(0); |
| __Ipnu = _Tp(0); |
| } |
| __Knu = std::numeric_limits<_Tp>::infinity(); |
| __Kpnu = -std::numeric_limits<_Tp>::infinity(); |
| return; |
| } |
| |
| const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); |
| const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon(); |
| const int __max_iter = 15000; |
| const _Tp __x_min = _Tp(2); |
| |
| const int __nl = static_cast<int>(__nu + _Tp(0.5L)); |
| |
| const _Tp __mu = __nu - __nl; |
| const _Tp __mu2 = __mu * __mu; |
| const _Tp __xi = _Tp(1) / __x; |
| const _Tp __xi2 = _Tp(2) * __xi; |
| _Tp __h = __nu * __xi; |
| if ( __h < __fp_min ) |
| __h = __fp_min; |
| _Tp __b = __xi2 * __nu; |
| _Tp __d = _Tp(0); |
| _Tp __c = __h; |
| int __i; |
| for ( __i = 1; __i <= __max_iter; ++__i ) |
| { |
| __b += __xi2; |
| __d = _Tp(1) / (__b + __d); |
| __c = __b + _Tp(1) / __c; |
| const _Tp __del = __c * __d; |
| __h *= __del; |
| if (std::abs(__del - _Tp(1)) < __eps) |
| break; |
| } |
| if (__i > __max_iter) |
| std::__throw_runtime_error(__N("Argument x too large " |
| "in __bessel_jn; " |
| "try asymptotic expansion.")); |
| _Tp __Inul = __fp_min; |
| _Tp __Ipnul = __h * __Inul; |
| _Tp __Inul1 = __Inul; |
| _Tp __Ipnu1 = __Ipnul; |
| _Tp __fact = __nu * __xi; |
| for (int __l = __nl; __l >= 1; --__l) |
| { |
| const _Tp __Inutemp = __fact * __Inul + __Ipnul; |
| __fact -= __xi; |
| __Ipnul = __fact * __Inutemp + __Inul; |
| __Inul = __Inutemp; |
| } |
| _Tp __f = __Ipnul / __Inul; |
| _Tp __Kmu, __Knu1; |
| if (__x < __x_min) |
| { |
| const _Tp __x2 = __x / _Tp(2); |
| const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; |
| const _Tp __fact = (std::abs(__pimu) < __eps |
| ? _Tp(1) : __pimu / std::sin(__pimu)); |
| _Tp __d = -std::log(__x2); |
| _Tp __e = __mu * __d; |
| const _Tp __fact2 = (std::abs(__e) < __eps |
| ? _Tp(1) : std::sinh(__e) / __e); |
| _Tp __gam1, __gam2, __gampl, __gammi; |
| __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); |
| _Tp __ff = __fact |
| * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); |
| _Tp __sum = __ff; |
| __e = std::exp(__e); |
| _Tp __p = __e / (_Tp(2) * __gampl); |
| _Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi); |
| _Tp __c = _Tp(1); |
| __d = __x2 * __x2; |
| _Tp __sum1 = __p; |
| int __i; |
| for (__i = 1; __i <= __max_iter; ++__i) |
| { |
| __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); |
| __c *= __d / __i; |
| __p /= __i - __mu; |
| __q /= __i + __mu; |
| const _Tp __del = __c * __ff; |
| __sum += __del; |
| const _Tp __del1 = __c * (__p - __i * __ff); |
| __sum1 += __del1; |
| if (std::abs(__del) < __eps * std::abs(__sum)) |
| break; |
| } |
| if (__i > __max_iter) |
| std::__throw_runtime_error(__N("Bessel k series failed to converge " |
| "in __bessel_jn.")); |
| __Kmu = __sum; |
| __Knu1 = __sum1 * __xi2; |
| } |
| else |
| { |
| _Tp __b = _Tp(2) * (_Tp(1) + __x); |
| _Tp __d = _Tp(1) / __b; |
| _Tp __delh = __d; |
| _Tp __h = __delh; |
| _Tp __q1 = _Tp(0); |
| _Tp __q2 = _Tp(1); |
| _Tp __a1 = _Tp(0.25L) - __mu2; |
| _Tp __q = __c = __a1; |
| _Tp __a = -__a1; |
| _Tp __s = _Tp(1) + __q * __delh; |
| int __i; |
| for (__i = 2; __i <= __max_iter; ++__i) |
| { |
| __a -= 2 * (__i - 1); |
| __c = -__a * __c / __i; |
| const _Tp __qnew = (__q1 - __b * __q2) / __a; |
| __q1 = __q2; |
| __q2 = __qnew; |
| __q += __c * __qnew; |
| __b += _Tp(2); |
| __d = _Tp(1) / (__b + __a * __d); |
| __delh = (__b * __d - _Tp(1)) * __delh; |
| __h += __delh; |
| const _Tp __dels = __q * __delh; |
| __s += __dels; |
| if ( std::abs(__dels / __s) < __eps ) |
| break; |
| } |
| if (__i > __max_iter) |
| std::__throw_runtime_error(__N("Steed's method failed " |
| "in __bessel_jn.")); |
| __h = __a1 * __h; |
| __Kmu = std::sqrt(__numeric_constants<_Tp>::__pi() / (_Tp(2) * __x)) |
| * std::exp(-__x) / __s; |
| __Knu1 = __Kmu * (__mu + __x + _Tp(0.5L) - __h) * __xi; |
| } |
| |
| _Tp __Kpmu = __mu * __xi * __Kmu - __Knu1; |
| _Tp __Inumu = __xi / (__f * __Kmu - __Kpmu); |
| __Inu = __Inumu * __Inul1 / __Inul; |
| __Ipnu = __Inumu * __Ipnu1 / __Inul; |
| for ( __i = 1; __i <= __nl; ++__i ) |
| { |
| const _Tp __Knutemp = (__mu + __i) * __xi2 * __Knu1 + __Kmu; |
| __Kmu = __Knu1; |
| __Knu1 = __Knutemp; |
| } |
| __Knu = __Kmu; |
| __Kpnu = __nu * __xi * __Kmu - __Knu1; |
| |
| return; |
| } |
| |
| |
| /** |
| * @brief Return the regular modified Bessel function of order |
| * \f$ \nu \f$: \f$ I_{\nu}(x) \f$. |
| * |
| * The regular modified cylindrical Bessel function is: |
| * @f[ |
| * I_{\nu}(x) = \sum_{k=0}^{\infty} |
| * \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} |
| * @f] |
| * |
| * @param __nu The order of the regular modified Bessel function. |
| * @param __x The argument of the regular modified Bessel function. |
| * @return The output regular modified Bessel function. |
| */ |
| template<typename _Tp> |
| _Tp |
| __cyl_bessel_i(const _Tp __nu, const _Tp __x) |
| { |
| if (__nu < _Tp(0) || __x < _Tp(0)) |
| std::__throw_domain_error(__N("Bad argument " |
| "in __cyl_bessel_i.")); |
| else if (__isnan(__nu) || __isnan(__x)) |
| return std::numeric_limits<_Tp>::quiet_NaN(); |
| else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) |
| return __cyl_bessel_ij_series(__nu, __x, +_Tp(1), 200); |
| else |
| { |
| _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; |
| __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); |
| return __I_nu; |
| } |
| } |
| |
| |
| /** |
| * @brief Return the irregular modified Bessel function |
| * \f$ K_{\nu}(x) \f$ of order \f$ \nu \f$. |
| * |
| * The irregular modified Bessel function is defined by: |
| * @f[ |
| * K_{\nu}(x) = \frac{\pi}{2} |
| * \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi} |
| * @f] |
| * where for integral \f$ \nu = n \f$ a limit is taken: |
| * \f$ lim_{\nu \to n} \f$. |
| * |
| * @param __nu The order of the irregular modified Bessel function. |
| * @param __x The argument of the irregular modified Bessel function. |
| * @return The output irregular modified Bessel function. |
| */ |
| template<typename _Tp> |
| _Tp |
| __cyl_bessel_k(const _Tp __nu, const _Tp __x) |
| { |
| if (__nu < _Tp(0) || __x < _Tp(0)) |
| std::__throw_domain_error(__N("Bad argument " |
| "in __cyl_bessel_k.")); |
| else if (__isnan(__nu) || __isnan(__x)) |
| return std::numeric_limits<_Tp>::quiet_NaN(); |
| else |
| { |
| _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; |
| __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); |
| return __K_nu; |
| } |
| } |
| |
| |
| /** |
| * @brief Compute the spherical modified Bessel functions |
| * @f$ i_n(x) @f$ and @f$ k_n(x) @f$ and their first |
| * derivatives @f$ i'_n(x) @f$ and @f$ k'_n(x) @f$ |
| * respectively. |
| * |
| * @param __n The order of the modified spherical Bessel function. |
| * @param __x The argument of the modified spherical Bessel function. |
| * @param __i_n The output regular modified spherical Bessel function. |
| * @param __k_n The output irregular modified spherical |
| * Bessel function. |
| * @param __ip_n The output derivative of the regular modified |
| * spherical Bessel function. |
| * @param __kp_n The output derivative of the irregular modified |
| * spherical Bessel function. |
| */ |
| template <typename _Tp> |
| void |
| __sph_bessel_ik(const unsigned int __n, const _Tp __x, |
| _Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n) |
| { |
| const _Tp __nu = _Tp(__n) + _Tp(0.5L); |
| |
| _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; |
| __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); |
| |
| const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2() |
| / std::sqrt(__x); |
| |
| __i_n = __factor * __I_nu; |
| __k_n = __factor * __K_nu; |
| __ip_n = __factor * __Ip_nu - __i_n / (_Tp(2) * __x); |
| __kp_n = __factor * __Kp_nu - __k_n / (_Tp(2) * __x); |
| |
| return; |
| } |
| |
| |
| /** |
| * @brief Compute the Airy functions |
| * @f$ Ai(x) @f$ and @f$ Bi(x) @f$ and their first |
| * derivatives @f$ Ai'(x) @f$ and @f$ Bi(x) @f$ |
| * respectively. |
| * |
| * @param __n The order of the Airy functions. |
| * @param __x The argument of the Airy functions. |
| * @param __i_n The output Airy function. |
| * @param __k_n The output Airy function. |
| * @param __ip_n The output derivative of the Airy function. |
| * @param __kp_n The output derivative of the Airy function. |
| */ |
| template <typename _Tp> |
| void |
| __airy(const _Tp __x, |
| _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip) |
| { |
| const _Tp __absx = std::abs(__x); |
| const _Tp __rootx = std::sqrt(__absx); |
| const _Tp __z = _Tp(2) * __absx * __rootx / _Tp(3); |
| |
| if (__isnan(__x)) |
| return std::numeric_limits<_Tp>::quiet_NaN(); |
| else if (__x > _Tp(0)) |
| { |
| _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; |
| |
| __bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); |
| __Ai = __rootx * __K_nu |
| / (__numeric_constants<_Tp>::__sqrt3() |
| * __numeric_constants<_Tp>::__pi()); |
| __Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi() |
| + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3()); |
| |
| __bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); |
| __Aip = -__x * __K_nu |
| / (__numeric_constants<_Tp>::__sqrt3() |
| * __numeric_constants<_Tp>::__pi()); |
| __Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi() |
| + _Tp(2) * __I_nu |
| / __numeric_constants<_Tp>::__sqrt3()); |
| } |
| else if (__x < _Tp(0)) |
| { |
| _Tp __J_nu, __Jp_nu, __N_nu, __Np_nu; |
| |
| __bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); |
| __Ai = __rootx * (__J_nu |
| - __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); |
| __Bi = -__rootx * (__N_nu |
| + __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); |
| |
| __bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); |
| __Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3() |
| + __J_nu) / _Tp(2); |
| __Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3() |
| - __N_nu) / _Tp(2); |
| } |
| else |
| { |
| // Reference: |
| // Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions. |
| // The number is Ai(0) = 3^{-2/3}/\Gamma(2/3). |
| __Ai = _Tp(0.35502805388781723926L); |
| __Bi = __Ai * __numeric_constants<_Tp>::__sqrt3(); |
| |
| // Reference: |
| // Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions. |
| // The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3). |
| __Aip = -_Tp(0.25881940379280679840L); |
| __Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3(); |
| } |
| |
| return; |
| } |
| |
| _GLIBCXX_END_NAMESPACE_VERSION |
| } // namespace std::tr1::__detail |
| } |
| } |
| |
| #endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC |
